Injective envelopes of partial C*-dynamical systems

We extend Hamana's theory of injective envelopes, along with several key features of the theory, to the realm of partial C*-dynamical systems.

In particular, we show that a partial C*-dynamical system has the ideal intersection property if and only if its injective envelope does.

A key ingredient in our arguments is a new kind of unitization of a partial action on a unital C*-algebra $A$ arising from the C*-algebra generated by the orbits of $A$ in its injective envelope~$I(A)$.

For an arbitrary unital partial C*-dynamical system, which is known to have an enveloping action, we establish a natural relationship between the injective envelope of the system and the injective envelope of its enveloping action.

For an abelian partial C*-dynamical system, we show that our construction coincides with the algebra of continuous functions on the Furstenberg boundary of the corresponding transformation groupoid.

It is crucial in our work to consider a notion of generalized unital partial C*-dynamical systems, in which the unital ideals are replaced by unital hereditary subalgebras.